Get outside wireframe
You can use Wireframe or Skin modifiers(note that you can adjust the radius per vertex in the n panel while in edit mode for skin modifier or with ctrl+a).

With Wireframe modifier you'll have to keep an eye on your mesh edges. Use N-gons wisely in order to pass the operator just the edges you need. For the provided example, a solution could be something like:



As you see the trick is to not build a complete mesh, but just the faces that hold the right edges. You'll also be able to assign different materials to the edges with vertex paint without having to apply the modifier.

You can also select only edges you want, invert selection with ctrl+i, delete everything else, select all and delete only faces (x menu) and then you are left with edges only. Then from object mode you can convert them to curves(alt+c) and then you can add thickness to them:



If you wish once you are happy with them you may convert them back to polygons with the same shortcut alt+c.